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Abstract

Background

Detection of significant differentially expressed genes (DEGs) from DNA microarray
datasets is a common routine task conducted in biomedical research. For the detection
of DEGs, numerous methods are proposed. By such conventional methods, generally, DEGs
are detected from one dataset consisting of group of control and treatment. However,
some DEGs are easily to be detected in any experimental condition. For the detection
of much experiment condition specific DEGs, each measurement value of gene expression
levels should be compared in two dimensional ways, or both with other genes and other
datasets simultaneously. For this purpose, we retrieve the gene expression data from
public database as possible and construct "meta-dataset" which summarize expression
change of all genes in various experimental condition. Herein, we propose "two-way
AIC" (Akaike Information Criteria), method for simultaneous detection of significance
genes and experiments on meta-dataset.

Results

As a case study of the Pseudomonas aeruginosa, we evaluate whether two-way AIC method can detect test data which is the experiment
condition specific DEGs. Operon genes are used as test data. Compared with other commonly
used statistical methods (t-rank/F-test, RankProducts and SAM), two-way AIC shows the highest specificity of
detection of operon genes.

Conclusions

The two-way AIC performs high specificity for operon gene detection on the microarray
meta-dataset. This method can also be applied to estimation of mutual gene interactions.

Background

Detection of significant differentially expressed genes (DEGs) from DNA microarray
datasets is a common routine task conducted in biomedical research [1-3]. For the detection of DEGs, numerous methods are proposed [4-7]. By such conventional methods, generally, DEGs are detected from one dataset consisting
of group of control and treatment. However, some DEGs are easily to be detected in
very wide or common experimental conditions. For example, "pyoverdin" genes (pvdD and pvdJ) [8] of Pseudomonas aeruginosa, which are ones of Iron transporter proteins and involved in cell division, are generally
detected as DEGs in experimental conditions which are conducted to observe cell division
(such as GSE24784 in GEO database) (Figure 1). Additionally, in analyses of some expression dataset of public database by commonly
used statistical methods, pyoverdin genes are also detected as DEGs in many other
experimental condition which are not conducted to observe cell division. Literatures
suggested that this may be because of pyoverdin is involved in many other biological
processes such as cell-to-cell signaling (Quorum Sensing, QS) [9] and virulence factor production [10]. In this way pyoverdin genes are prone to be detected as DEGs in any experiment condition,
however, many researchers may want to these genes to be detected in the special experiments
(i.e., cell division condition). For this purpose, each measurement value of gene
expression levels should be compared in two dimensional ways, or both with other genes
and other datasets simultaneously.

Figure 1.Expression change of pyoverdin genes. We analyze some expression data of pyoverdin genes (pvdD and pvdJ) of public database (GEO and Array- Express) by commonly used statistical methods
(log-FC, RankProducts, t-rank and SAM). The threshold value of log-FC is set to 2 (4-fold) and that of RankProducts,
t-rank and SAM are set to upper 300 gene. All dataset are normalized by RMA method
separately. If both genes are co-expressed, corresponding box is filled in white,
otherwise gray. Figure shows that pyoverdin genes are prone to be detected in any
experiment condition and our method focuses on much experiment condition specific
DEGs (GSE7704).

For the detection of such DEGs, we retrieve the gene expression data from public database
as possible and construct "meta-dataset" which summarize expression change of all
genes in various experiment condition (Figure 2). Although there are no 'de fact' standard definition for meta-datasets, log ratio
value which are widely used to analyze DNA microarray data can be introduced to construct
meta-datasets when each dataset is consist of control and treatment experiment data.

Figure 2.Meta-dataset and log-FC matrix. A meta-dataset is a set of multiple datasets. Each dataset consists of a control
group and a treatment group, each of which has one or more DNA microarray data. The
measured probe (gene) is common to all datasets. The element of Fi,j in log-FC matrix is the log-transformed (base 2) fraction of arithmetic mean values
of treatment and control group in i-th gene of j-th dataset.

In such meta-datasets, direct application of widely used conventional statistical
methods is not suitable to detect two-dimensional DEGs because such methods are intended
to find special genes among all experiments to be analyzed.

For example, ANOVA [11-14] is applied very widely for multi-group analysis method, but its concludes only that
differences between groups (genes) are significant or not. Therefore ANOVA can not
detect simultaneously specific genes in specific experiments as two-dimensional DEGs.

Outlier detection methods are also widely used to detect DEGs, such as Shannon entropy
[15] or Sprent's non-parametric method [16]. In difference to ANOVA, these methods can also detect both special genes or special
experimental conditions, but it is not simultaneously. It is one-dimensional and similar
to ANOVA.

Multiple testing [17] (multiple comparisons, such as Bonferroni correction, Tukey-Kramer's method, and
Games-Howell's method) also produce limited results as same as outlier detections.
For an example of a dataset consisting of N genes and E experiments, it never means that the i-th gene of the j-th experiment is a DEG when multiple testing shows that the i-th gene (size E vector) is significantly different from other genes and the j-th experiment (size N vector) is significantly different from other experiments independently. This is because
most multiple testing methods are conducted to ascertain differences between mean
values of groups.

Herein, we propose "two-way AIC" (Akaike Information Criteria) method for simultaneous
detection of significant genes and experiments on metadatasets. This method detects
specific genes that are differentially expressed in specific experimental conditions.
Here, we present comparison of the performance of our method to other widely used
statistical methods and show that two-way AIC method has high specificity for detection
of test data which tend to express in specific experiment condition.

Methods

Meta-dataset and log-FC matrix

A meta-dataset is a set of plural datasets. Each dataset consists of measurement groups
of two kinds: control and treatment. Both control and treatment groups consist of
one or more DNA microarray measurements. Genes (probes) are common to all microarrays
(Figure 2).

After normalization is applied, we summarized the expression data of each dataset
as logarithm of fold change values (log-FC). This step is for removal of systematic
bias between samples of different studies [18]. Log-FC of each gene are calculated based on ratios of measurement values of treatment
to those of control for each dataset. Log-FC is defined as a logarithm (base 2) of
a fraction of arithmetic mean values of treatment and control shown as follows:

(1)

where and respectively denote the arithmetic mean values of treatment and control measurements
of i-th gene of j-th dataset (Figure 2). We define the row side direction of the matrix of log-FC values (log-FC matrix)
as the "gene side" and the column side direction as the "experiment side".

Judgment matrix

Here we define the judgment matrix, which is the conclusion based on results of DEG
detections described as a two-dimensional table (gene and experiment) (Figure 3). The element xi,j in the judgment matrix is the result of DEG detection of the i-th gene in the j-th experiment (dataset). Each element takes one value out of three values: 1, -1,
or 0.1 means positive DEG (specifically higher expression), -1 means negative DEG
(specifically lower expression) and 0 means that it is not a DEG. Generally, DEG detection
can be performed both gene side and experiment side direction.

Figure 3.Judgment matrix. The judgment matrix is the summary of results of each DEG detection method. This
matrix is derived from the meta-dataset or log-FC matrix, where each element has one
value: 1 (positive DEG), -1 (negative DEG) or 0 (non-DEG).

Two-way AIC

Our two-way AIC, based on the U-value method [19,20], is applied to the log-FC matrix. It detects DEGs as outliers of both the gene side
and the experiment side simultaneously. Given a group of samples, and the n furthest samples from the group's average are presumed as outliers, the U-value is defined as

(2)

where n is the number of outliers, and σ and s respectively denote the standard deviation and the number of non-outlier samples.
Outliers are estimated as the best presumption of outliers which minimizes U. In this paper, the search range is restricted to within 25 percent of the number
of data.

When the U-value method is applied in the gene side direction, specific experiments are detected
as outliers for each gene. Similarly, when the U-value method is applied in the experiment side, specific genes are detected for each
experiment. The detected outliers are described as 1 (positive outlier) or -1 (negative
outlier).

Detection results of i-th gene of j-th experiment have two labels, the result on the gene side and that of experiment
side direction. xi,j in the judgment matrix is set to the value of the label if two labels are the same.
Finally it is judged as a DEG (Figure 4). The element (xi,j) of the judgment matrix of two-way AIC is described as

(3)

where is the element on the i-th gene, j-th experiment in the judgment matrix by Ueda's statistic on the experiment side and
is the element on the i-th gene, j-th experiment in the judgment matrix by Ueda's statistic on the gene side.

Figure 4.DEGs in two-way AIC. Two-way AIC detects DEGs as outliers of the gene side and the experiment side. In
each direction, outliers are detected by U-value method. Outliers are described as 1 (positive outlier) or -1 (negative outlier).
Based on two sets of results for each direction, xi,j in the judgment matrix is set to the value of the label if two labels are the same.
Finally it is judged as a DEG (Figure 3).

Results

The two-way AIC method is applied to a prokaryote gene expression meta-dataset to
demonstrate its detection performance, and it is compared in specificity of detection
of test data (operon genes) [21,22], which generally tend to express simultaneously against specific experiment condition
with other widely used statistical methods.

Data

A meta-dataset is set up by calculating the log-FC matrix from P.aeruginosa DNA microarray measurements diverse experimental conditions. DNA microarray datasets
are retrieved from two public databases: the Gene Expression Omnibus (GEO) [23] and the ArrayExpress [24]. The measurement platform is the Affymetrix GeneChip® Pseudomonas aeruginosa Genome Array (registered as GPL84 in GEO and A-AFFY-30 in ArrayExpress),
which consists of 5883 probes (5549 protein coding genes of the PAO1 strain, 18 tRNA
and rRNA of the PAO1, 117 genes from other strains and 199 intergenic sequences).
We extract 5549 coding genes from 289 datasets (282 from GEO and 7 from Array- Express),
which do not contain Null values (NA or missing values) or 0. RMA normalization [25] is applied to the microarray datasets in each study. Then the log-FC matrix is calculated.

Operon genes

We use test data for evaluation of our method. Here we assess the method's performance
of detection of data which should be detected and evaluate its selectivity. We focus
on the operon gene, one of the biological mechanism. Operon genes which prokaryote
originally have are transcripted at same time and correspond to common function [26,27]. Therefore, we think these genes must be co-expressed against specific experiment
condition because of necessity of functional expression. We identify 93 operon genes
in 5549 codings genes by Operon Database [28] at Kyoto University and the Pseudomonas Genome Database [29] at the University of British Columbia. When a pair of two genes is chosen from an
operon, the number of all possible gene pairs is 857 for these 93 operons. Actually,
Pearson's correlation coefficient of these 857 operon gene pairs is 0.734 and shows
strong positive correlation, whereas that of randomly chosen gene pairs is 0.182 on
the log-FC matrix. Therefore, we use operon gene as objective test data. Operon genes
are not necessary to be expressed in any experimental condition. However, once some
genes which belong to an operon, all the operon genes should be expressed simultaneously.
Therefore, we regard operon genes which changed its expression level in specific experimental
condition as correct data in the experiment condition and non-operon genes as incorrect
data. Here we compare all method by evaluating how specifically detect these operon
genes.

Compared methods

We compare our two-way AIC method to other widely used DEG detection methods; t-rank [30] with F-test (experiment side in meta-dataset), RankProducts [31] (experiment side in meta-dataset), SAM (significance analysis of microarray) [32] (experiment side in meta-dataset), one side U- value outlier detection [19] (both gene side and experiment side in log-FC matrix), 2-σ (both sides simultaneously in log-FC matrix) and 3-σ (both sides simultaneously in log-FC matrix) (Table 1).

The judgment criterion of the t-rank with F-test, the RankProducts method and SAM is set to the rank which makes the sensitivity
of these methods closest to that of the two-way AIC. In the F-test, we evaluate the equality of variance (p = 0.05), and in the case of equal variances, we calculate Student's t-statistic, otherwise Welch's t-statistic with the threshold value (upper 245 genes). The RankProducts method is
a non-parametric FC based DEG detection method. We used it with the threshold value
(upper 312 genes). SAM is a non-parametric t-statistic based DEG detection method. We used it with the threshold value (upper
96 genes).

In the 2- and 3-σ methods, log-FC values of genes that are larger than the threshold in both sides are
detected as DEGs. The threshold is the standard deviation multiplied by 2 (2σ method) and 3 (3σ method). σ is calculated for each direction.

Analyses of detected genes

The expected DEGs of each dataset in the meta- dataset mutually differ because their
experimental conditions differ. Therefore we report the detection performances of
the two-way AIC and other methods to show how precisely operon genes are detected
simultaneously. For all pairs of detected genes (denoted by gene a and b) as DEGs by each detection method, then the pair is a "detected operon gene pairs"
when there is j in the judgment matrix so that xa,j = xb,j ≠ 0. Performance, sensitivity, specificity, p-value, the number and the percentage of DEGs are calculated as follows:

(4)

(5)

(6)

(7)

(8)

where N is the number of operons in which the belonging genes were detected as DEGs at least
once (0 ≤ N ≤ 93), M is the number of experiments in which belonging genes were detected as DEGs at least
once (0 ≤ M ≤ 289), Ok,j is the number of detected operon gene pairs, Tk is the number of all possible operon gene pairs in k-th operon, Ak,j is the number of never-detected non-operon gene pairs, Pk,j is the p-value in the k-th operon, j-th experiment calculated using Fisher's exact test, F is the number of all possible combination of non-operon gene pairs (5549C2 - 857 = 15392069), G is the total number of genes (5549), E is the total number of all experimental conditions (289), and nj is the number of DEGs in the j-th experiment.

Scalability

Scalability of two-way AIC is assessed by some square matrices of random numbers (Figure
5). The x-axis shows the number of rows (or columns) of the square matrix. The y-axis
is computation time in minutes necessary to finish the calculation. The linear regression
model by the least squares method is y = 8.30×10-6 · x2.47, where the coefficient of determination is 0.9946. Therefore, the calculation cost
of the two-way AIC is estimated to be polynomial: O(x2.47). Computational time is measured using GNU R 2.15.0 on Mac OS × 10.6.8, 2.4 GHz Intel
Core 2 Duo, and 8 GB 1067 MHz DDR3 RAM.

Figure 5.Scalability of two-way AIC. Scalability is assessed using some square matrices. Each element of a square matrix
is set to a pseudorandom number of a normal distribution. The x-axis is the number
of row (or column) of the matrix and the y-axis is the computation time in minutes
necessary to finish the calculation. Both axes are transformed as logarithmic values
(base 10).

Discussion

Results show that the two-way AIC is superior to all other method in p-value and specificity. It means that false positives of the two-way AIC is the lowest.
Among other widely used methods (t-rank/F-test, RankProducts and SAM), SAM shows the highest specificity. However, specificity
of our method is much higher than that of SAM. It suggest the effectiveness of two-way
approach. Compared with other two-way method (2-σ, 3-σ), specificity of two- way AIC is also highest. It means specificity of U-value is superior to that of standard deviation in this case. Therefore, the two-way
AIC method can detect operon genes with less noises even with all genes in an operon
do not alway express proportionally [33].

Detection sensitivity is generally lower compared for specificity of all methods we
tested. Compared to U-value method (gene side and experiment side), sensitivity of two-way AIC is not high.
In general, one-way methods (U-value methods in Table 1) detects more operon genes than two-way methods because these methods are considered
as one-pass outlier filtering while two-way methods are double filtering. However
result show that double filtering cause much low false positive and choose genes that
should be detected.

Any statistic including the t-test can be applied in two-way approach to meta-datasets in general, however, how
to set the detection criterion or threshold of outliers is a major concern in these
approaches. Introducing a model selection criteria AIC does not needed trial and error
to find optimal threshold.

The stability of detection methods is shown in Figure 6. Significance level based methods (Welch's t-test, Benjamini-Hochberg method (BH) method [34] and Wilcoxon rank sum test often show anomalous results in which most DEGs are found
in a few measurements. In the case of the Wilcoxon test, large numbers of DEGs are
detected for a few experimental conditions and almost nothing is found for many conditions,
and its detection results are highly variable depending on detection criteria (p-values of 0.05 to 0.001). It can be almost meaningless to detect DEGs from a meta-dataset
that includes a wide variety of experimental conditions. Larger p-value or q-value is needed for test criteria to improve such detection of Welch's t-test and BH method, however, such large threshold will allow to result detecting
extremely a large number of DEGs in a specific few experiments. For example, about
3000 genes are detected in Welch t-test with 0.05 p-value. Analyzing of multiple dataset uniformly by single significance level is difficult.
Such situation is also found other meta-analysis study [35]. Steepness of the curve by the two-way AIC is milder than those of these methods,
which means that it is less anomalous.

Figure 6.Number of DEGs in each experiment detected by significance level. We performed Welch t-test (A), BH (Benjamini-Hochberg) method with p-value of Welch t-test (B) and Wilcoxon rank sum test (C) against 209 datasets with four significance
level (0.001, 0.005, 0.01, 0.05). The 209 datasets, chosen from 289 datasets in this
paper consist of control and treatment groups, each of which has two or more samples
so that we can calculate the variance. The x-axis is the dataset sorted by the number
of DEGs. The y-axis is the number of DEGs in each dataset. Performance of our two-way
AIC is shown as (D).

Finally, we show an application of our two-way AIC method to detecting mutual gene
interactions. lasI, which is one of the QS-related gene, is suggested to regulate biofilm formation
[36]. Biofilm is the mucoidy structure consisting of polysaccharide that bacteria produced.
QS intervention against Biofilm formation is phenotypically observed by mutation experiment.
However, its biological mechanisms such as pathway, gene regulation, molecular mechanism
or other specific molecular biological evidence is still unknown [37,38]. In the judgement matrix of two-way AIC, this interaction is actually observed in
two experiment condition (Figure 7) and these condition is designed by two independent researches. Both researches used
P.aeruginosa which is isolated from Cystic Fibrosis Patients [39,40]. Actually biofilm contributes some diseases [41] and especially relationship of Cystic Fibrosis [42] is attracting attention of many researchers [43]. Interestingly, QS intervention to biofilm is not mentioned in these literatures
because it is not a purpose of their experiments. However, the two-way AIC method
detects a possible gene interaction which implies that lasI is related to biofilm formation in Cystic Fibrosis patient and perhaps lasI inhibition will stop biofilm formation and Cystic Fibrosis. In this way two-way AIC
can help building hypothesis about mutual gene interaction across the multiple experimental
condition datasets.

Figure 7.Co-expression of lasI and alg operon. lasI, one of Quorum Sensing (QS) -releated gene (A) and alg operon (algD/alg8/alg44/algK /algE/algG/algX/algL/algI/algJ/algF/algA), biofilm-related genes (B) are detected by two-way AIC respectively. In the judgement
matrix of two-way AIC, these two kind of genes are co-expressed in GSE21966 and GSE9704
(C). It is suggested that these genes are related in two experiment condition.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

KT designed the study, retrieved all data used in this work, performed the analysis,
and drafted the manuscript. DT helped to design the study, to select statistical methods
to be compared, to interpret the result, and to draft the manuscript. YK and SM supervised
all work. All authors were involved in drafting the manuscript. They have read and
approved the final manuscript.

Declarations

The publication costs for this article were funded by the corresponding author's institution.